The idea is, the Moon is going in (approximately) a circle around the Earth, while the Earth is going in (approximately) a circle around the Sun. This is a surprisingly simple situation to model using a parametric equation:
The general form for this type of curve is
We can relate this to the earlier equations with
[Edit: If you're curious about the b > a/c condition, you can think of it this way – There will be a loop anytime the Moon is moving backwards faster than the Earth is moving forwards. The speed of each is given by 2πR/T, and if you plug in the R and T relations for a, b, and c, you'll see that's exactly what b > a/c means.]
Orion suggested that I post my non-formulaic solution. I'll give away the punch line - the important piece of data is that the earth and moon are close together compared with the earth and sun.
ReplyDeleteI don't have a graphic app to illustrate, but if you draw this yourself it will likely make more sense.
1. The number of degrees in a circle is close enough to the number of days in the year, that we can say the earth travels one degree a day. So draw a 30 degree arc of a big circle to represent the earth's path in a month.
2. Put 8-9 dots on the arc to represent evenly spaced locations in the earth's orbit.
3. With a contrasting color, put a dot near each earth to represent the moon. The first and last should have the moon in the same spot to represent the full cycle is complete.
4. Connect the contrasting color dots, and voila, you have the slightly wavy circle that Orion showed.